Ch. 4 - Discrete Probability Distributions

Larson - Elementary Statistics: Picturing the World 8th Edition

Larson8th EditionElementary Statistics: Picturing the WorldISBN: 9780137493470Not the one you use?Change textbook

Larson 8th Edition

Ch. 4 - Discrete Probability Distributions

Problem 4.1.40

Chapter 4, Problem 4.1.40

Baseball There were 116 World Series from 1903 to 2020. Use the probability distribution in Exercise 30 to find the number of World Series that had 4, 5, 6, 7, and 8 games. Find the population mean, variance, and standard deviation of the data using the traditional definitions. Compare to your answers in Exercise 30.

Verified step by step guidance

Step 1: Identify the probability distribution provided in Exercise 30. This distribution will give the probabilities for the number of games (4, 5, 6, 7, and 8) in a World Series. Denote the number of games as X and the corresponding probabilities as P(X).

Step 2: To find the expected number of World Series for each game count (4, 5, 6, 7, and 8), multiply the total number of World Series (116) by the probability of each game count. For example, for 4 games, calculate 116 × P(4). Repeat this for 5, 6, 7, and 8 games.

Step 3: Calculate the population mean (μ) using the formula for the expected value of a discrete random variable: μ = Σ[X × P(X)]. Multiply each game count (X) by its probability (P(X)) and sum the results.

Step 4: Calculate the population variance (σ²) using the formula: σ² = Σ[(X - μ)² × P(X)]. For each game count (X), subtract the mean (μ), square the result, multiply by the probability (P(X)), and sum these values.

Step 5: Calculate the population standard deviation (σ) by taking the square root of the variance: σ = √(σ²). Compare the calculated mean, variance, and standard deviation to the results from Exercise 30 to identify any differences or similarities.

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.

Video duration:

Was this helpful?

Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Probability Distribution

A probability distribution describes how the probabilities of a random variable are distributed across its possible values. In the context of the World Series, it would detail the likelihood of each series lasting a certain number of games (4, 5, 6, 7, or 8). Understanding this distribution is crucial for calculating expected values and other statistical measures.

Population Mean

The population mean is the average of a set of values, calculated by summing all the values and dividing by the total number of values. In this case, it represents the average number of games played in the World Series over the specified years. This measure provides insight into the typical length of a World Series.

Variance and Standard Deviation

Variance measures the spread of a set of values around the mean, indicating how much the values differ from the average. Standard deviation, the square root of variance, provides a measure of dispersion in the same units as the data. Both metrics are essential for understanding the variability in the number of games played in the World Series.

Recommended video:

Guided course

08:45

Calculating Standard Deviation

Related Practice

Textbook Question

"Using a Distribution to Find Probabilities In Exercises 11–26, find the indicated probabilities using the geometric distribution, the Poisson distribution, or the binomial distribution. Then determine whether the events are unusual. If convenient, use a table or technology to find the probabilities.

Typographical Errors A newspaper finds that the mean number of typographical errors per page is four. Find the probability that the number of typographical errors found on any given page is (a) exactly three, (b) at most three, and (c) more than three."

Textbook Question

Geometric Distribution: Mean and Variance In Exercises 29 and 30, use the fact that the mean of a geometric distribution is μ = 1/p and the variance is

sigma^2 = q/p^2

Paycheck Errors A company assumes that 0.5% of the paychecks for a year were calculated incorrectly. The company has 200 employees and examines the payroll records from one month. (a) Find the mean, variance, and standard deviation. (b) How many employee payroll records would you expect to examine before finding one with an error?

Textbook Question

Finding an Expected Value In Exercises 37 and 38, find the expected value E(x) to the player for one play of the game. If x is the gain to a player in a game of chance, then E(x) is usually negative. This value gives the average amount per game the player can expect to lose.

A high school basketball team is selling $10 raffle tickets as part of a fund-raising program. The first prize is a trip to the Bahamas valued at $5460, and the second prize is a weekend ski package valued at \$496. The remaining 18 prizes are \$100 gas cards. The number of tickets sold is 3500.

Textbook Question

In Exercises 1–4, find the indicated probability using the geometric distribution.

Find P(5) when p = 0.09

Textbook Question

Finding Binomial Probabilities In Exercises 19–26, find the indicated probabilities. If convenient, use technology or Table 2 in Appendix B.

Penalty Kicks Argentine soccer player Lionel Messi converts 78% of his penalty kicks. Suppose Messi takes six penalty kicks next season. Find the probability that the number he converts is (a) exactly six, (b) at most three, and (c) more than three. (Source: Transfermarkt)

views

Textbook Question

True or False? In Exercises 5–8, determine whether the statement is true or false. If it is false, rewrite it as a true statement.

For a random variable x, the word random indicates that the value of x is determined by chance.